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Deterministic versus Probabilistic

Deterministic: All data is known beforehand Once you start the system, you know exactly what is going to happen. Example. Predicting the amount of money in a bank account.If you know the initial deposit, and the interest rate, then: You can determine the amount in the account after one year.

Probabilistic: Element of chance is involved You know the likelihood that something will happen, but you dont know when it will happen. Example. Roll a die until it comes up 5.Know that in each roll, a 5 will come up with probability 1/6. Dont know exactly when, but we can predict well.

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Basic ProbabilityDenition: An experiment is any process whose outcome is uncertain. Denition: The set of all possible outcomes of an experiment is called the sample space, denoted X or S . Denition: Each outcome x X has a number between 0 and 1 that measures its likelihood of occurring. This is the probability of x , denoted p (x ). Example. Rolling a die is an experiment; the sample space is }. The individual probabilities are all p (i ) = . { Denition: An event E is something that happens (in other words, a subset of the sample space). Denition: Given E , the probability of the event (p (E )) is the sum of the probabilities of the outcomes making up the event. Example. The roll of the die . . . [is 5] or [is odd] or [is prime] . . . , p (E2 ) = , p (E3 ) = . Example. p (E1 ) =

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Determining ProbabilitiesThree methods for determining the probability of an occurrence: Relative frequency method: Repeat an experiment many occurrences times; assign as the probability the fraction # experiments run . Example. Hit a bulls-eye 17 times out of 100; set the probability of hitting a bulls-eye to be p (bulls-eye) = 0.17. Equal probability method: Assume all outcomes have 1 equal probability; assign as the probability # of possible outcomes . Example. Each side of a dodecahedral die is equally likely to 1 . appear; decide to set p (1) = 12 Subjective guess method: If neither method above applies, give it your best guess. Example. How likely is it that your friend will come to a party?

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Independent EventsDenition: Two events are independent if the probabilities of occurrence do not depend on one another. Example. Roll a red die and a blue die. Event 1: blue die rolls a 1. Event 2: red die rolls a 6. These events are independent. Event 1: blue die rolls a 1. Event 2: blue die rolls a 6. These events are dependent. Example. Pick a card, any card! Shue a deck of 52 cards. Event 1: Pick a rst card. Event 2: Pick a second card. . These events are Example. You wake up and dont know what day it is. Event 1: Today is a weekday. Event 2: Today is cloudy. Event 3: Today is Modeling day. E1 vs. E2 E2 vs. E3 E1 vs. E3

Basic Probability 5.3A (pp. 377391)

Decision TreesDenition: A multistage experiment is one in which each stage is a simpler experiment. They can be represented using a tree diagram. Each branch of the tree represents one outcome x of that levels experiment, and is labeled by p (x ). Example. Flipping a biased coin twice. 2/3 HH 2/3 H HT 1/3 2/3 1/3 T 1/3 TH TT Example. Indiana and SF State U. play two soccer games. (p. 382) 3/4 2: Ind 3 8 1 1: Ind 2 1/4 2: SF 1 81 2

4 9 2 9 2 9 1 9

1/3 1: SF 2/3

2: Ind 2: SF

1 6 1 3

Independent or dependent?

Independent or dependent?

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Expected value / mean

Even with the randomness, what do you expect to happen? Suppose that each outcome in a sample space has a number r (x ) attached to it. (examples: number of pips on a die, amount of money you win on a bet, inches of precipitation falling) This function r is called a random variable. Denition: The expected value or mean of a random variable is the sum of the numbers weighted by their probabilities. Mathematically, = E[X ] = p (x1 )r (x1 ) + p (x2 )r (x2 ) + + p (xn )r (xn ). Idea: With probability p (x1 ), there is a contribution of r (x1 ), etc. Example. How many heads would you expect on average when ipping a biased coin twice? Example. How many wins do you expect Indiana to have?

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Expected value / mean

When two random variables are on two independent experiments, the expected value operation behaves nicely: E[X + Y ] = E[X ] + E[Y ] and E[XY ] = E[X ]E[Y ]. Example. We throw a red die and a blue die. What is the expected value of the sum of the dice and the product of the dice?b+ r

1 2 3 4 5 6

1 2 3 4 5 6 7

2 3 4 5 6 7 8

3 4 5 6 4 5 6 7 5 6 7 8 6 7 8 9 7 8 9 10 8 9 10 11 9 10 11 12

r b

1 2 3 4 5 6

1 2 3 1 2 3 2 4 6 3 6 9 4 8 12 5 10 15 6 12 18

4 4 8 12 16 20 24

5 5 10 15 20 25 30

6 6 12 18 24 30 36

E[X + Y ] = E[XY ] =

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Component ReliabilityMany systems consist of components pieced together. To determine how reliable the system is, determine how reliable each component is and apply probability rules. Denition: The reliability of a system is its probability of success. Example. Launch the space shuttle into space with a three-stage rocket. Stage 1 Stage 2 Stage 3 In order for the rocket to launch, Let R1 = 90%, R2 = 95%, R3 = 96% be the reliabilities of Stages 13. p (system success) = p (S1 success and S2 success and S3 success)

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Component ReliabilityExample. Communicating with the space shuttle. There are two independent methods in which earth can communicate with the space shuttle A microwave radio with reliability R1 = 0.95 An FM radio, with reliability R2 = 0.96. In order to be able to communicate with the shuttle, . p (system success) = p (MW radio success or FM radio success)